Properties

Label 2-2016-252.247-c0-0-0
Degree $2$
Conductor $2016$
Sign $0.853 + 0.520i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.499 + 0.866i)21-s + (1.73 − i)23-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.5 − 0.866i)29-s + i·31-s + (0.5 − 0.866i)37-s + (0.866 − 0.499i)39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.499 + 0.866i)21-s + (1.73 − i)23-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.5 − 0.866i)29-s + i·31-s + (0.5 − 0.866i)37-s + (0.866 − 0.499i)39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.853 + 0.520i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.853 + 0.520i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7937448877\)
\(L(\frac12)\) \(\approx\) \(0.7937448877\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.395794327973724572542950809297, −8.428173491072348844807842524811, −7.44574116559945077173051130179, −6.85352636244171981249964815409, −6.27078246370046160692075894621, −5.34061011690685841305512665707, −4.49175921636616920517490711343, −3.49257383848948868118081263081, −2.24406348362199057281227418974, −0.888242199047591667072394774817, 0.984191250969131093375267666571, 2.95700914505775444901886205395, 3.39310884677969315616905081443, 4.94403320981449114731037742044, 5.24721955480935101847446320199, 6.11958908952040342168920661993, 7.05400669287435484507416342560, 7.59063404715677768496942707807, 9.003120812012892120012643006380, 9.516412455389990836658896422534

Graph of the $Z$-function along the critical line